Beamer: Reserving space for video overlay—automatically

Question

This question is rather similar to Beamer: Reserving space for video overlay :

For preparing a video lecture, I would like to have a frame template on which I can reserve a rectangular area, on which I want to place later a camera overlay for screencasting.

But with the additional requirement: I don't want to take care of not using the reserved space manually.

I tried to use \logo, but that does not behave as I had hoped. (See "cursus", footnote text, and caption in MWE.)

---

MWE:

\documentclass[english,beameralt,aspectratio=169]{beamer}
\usepackage{lipsum}
\usepackage{booktabs}
\usepackage[backend=bibtex]{biblatex}
\addbibresource{lipsum.bib}
\begin{filecontents}[overwrite]{lipsum.bib}
@misc{lipsum, author={\lipsum[8][1]}, title = {\lipsum[8][2]}}
\end{filecontents}
\renewcommand{\footfullcite}[1]{\cite{#1}\footnote[frame]{\fullcite{#1}}}
\logo{\textcolor{gray}{\rule{2cm}{2cm}}}
\begin{document}
\begin{frame}{\lipsum[1][1]}
    \begin{columns}
        \begin{column}{.5\textwidth}
            \lipsum[4]~\footfullcite{lipsum}
        \end{column}
        \begin{column}{.5\textwidth}
            \lipsum[2]
        \end{column}
    \end{columns}
\end{frame}
\begin{frame}{\lipsum[5][1]}
    \begin{columns}
        \begin{column}{.5\textwidth}
            \lipsum[6]
        \end{column}
        \begin{column}{.5\textwidth}
            \lipsum[7][1-8]
            \vspace*{\fill}
            \begin{table}
            \begin{tabular}{lrrrr}
                \toprule
                Lorem & Ipsum & Dolor & Sit & Amet\
                \midrule
                Lorem & Ipsum & Dolor & Sit & Amet\
                Lorem & Ipsum & Dolor & Sit & Amet\
                \bottomrule
            \end{tabular}
            \caption{\lipsum[3][1]}
            \end{table}
        \end{column}
    \end{columns}
\end{frame}
\end{document}

Result: